To find final velocity in physics, you usually pick one of a few standard formulas depending on what you’re given (time, acceleration, distance, etc.).

Core idea

Velocity changes when there is acceleration.
If acceleration is constant, you can relate initial velocity, final velocity, time, and displacement with kinematic equations.

Main formulas for final velocity

1. Using acceleration and time

Use this when you know:

  • Initial velocity viv_ivi​
  • Constant acceleration aaa
  • Time interval ttt

Formula:

vf=vi+aΔtv_f=v_i+a\Delta tvf​=vi​+aΔt

This means: start with the initial velocity and add “acceleration × time” to get final velocity.

Example idea:
A car starts at 10textm/s10\\text{m/s}10textm/s and accelerates at 2textm/s22\\text{m/s}^22textm/s2 for 5texts5\\text{s}5texts.

vf=10+2⋅5=20textm/sv_f=10+2\cdot 5=20\\text{m/s}vf​=10+2⋅5=20textm/s

2. Using distance (displacement)

Use this when you know:

  • Initial velocity viv_ivi​
  • Constant acceleration aaa
  • Displacement ddd (how far it traveled in the direction of motion)

Formula:

vf2=vi2+2adv_f^2=v_i^2+2advf2​=vi2​+2ad

Then take the square root to get vfv_fvf​.

Example idea:
If a ball starts at 0textm/s0\\text{m/s}0textm/s, accelerates at 3textm/s23\\text{m/s}^23textm/s2, and moves 10textm10\\text{m}10textm:

v_f^2=0^2+2\cdot 3\cdot 10=60 \Rightarrowv_f=\sqrt{60}\\text{m/s}

3. Using work–energy (forces and energy)

If you know how much work was done on an object, you can use energy.

  • Kinetic energy: KE=\tfrac{1}{2}mv^2
  • Work–energy theorem:

W=\Delta KE=KE_f-KE_i

Steps:

  1. Find initial kinetic energy: KE_i=\tfrac{1}{2}mv_i^2.
  2. Add work: KE_f=KE_i+W.
  3. Set KE_f=\tfrac{1}{2}mv_f^2 and solve for v_f.

This method is especially useful when you know forces and distances instead of time.

Quick strategy guide

When you’re given…

  • Initial velocity, acceleration, time → use v_f=v_i+at.
  • Initial velocity, acceleration, distance → use v_f^2=v_i^2+2ad.
  • Mass, forces, distances (work) → use energy/work–energy.

Think of it like picking the “equation that matches your knowns,” then solving for v_f.

Tiny story-style example

Imagine a skateboarder rolling at 4\\text{m/s} who goes down a ramp that gives a steady acceleration of 1.5\\text{m/s}^2 for 6\\text{s}.
You’d use v_f=v_i+at:

v_f=4+1.5\cdot 6=13\\text{m/s}

In those few seconds, her speed climbs from a gentle roll to a much faster glide, all captured by that simple formula.

HTML table of key formulas

html

<table>
  <thead>
    <tr>
      <th>Given</th>
      <th>Formula for final velocity</th>
      <th>Notes</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Initial velocity, acceleration, time</td>
      <td>v_f = v_i + a t</td>
      <td>Most common constant-acceleration equation. [web:5][web:9]</td>
    </tr>
    <tr>
      <td>Initial velocity, acceleration, displacement</td>
      <td>v_f^2 = v_i^2 + 2 a d</td>
      <td>Use when distance is known instead of time. [web:3][web:5]</td>
    </tr>
    <tr>
      <td>Mass, work done, initial velocity</td>
      <td>Use W = ΔKE and KE = ½ m v²</td>
      <td>Energy/work–energy approach, useful with forces & distances. [web:1][web:5]</td>
    </tr>
  </tbody>
</table>

TL;DR:
Most of the time, to find final velocity you use v_f=v_i+at or v_f^2=v_i^2+2ad, choosing the one that matches the information you’re given.